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Optimization problem involving an area and perimeter

  • #1
My friend and I have come across this problem in Apostol's Calculus Vol. 1, ed.2 (exercises 4.21 if anyone is looking). We are studying calculus independently and have become stumped by this one.

Homework Statement



The problem as written in Apostol:
"A farmer wishes to enclose a rectangular pasture of area A adjacent to a long stone wall. What dimensions require the least amount of fencing?"


Homework Equations


From the problem setup, we have deduced two elementary equations:

P = 2x + y
A = xy

where P is the perimeter (or length of fence, minus the stone wall part) and A is area (a constant).


The Attempt at a Solution



There are two equations here and two variables. That seemed like a good thing. We found y = A/x and then substituted into the perimeter equation: P(x) = 2x + A/x.

In order to optimize, we differentiated P(x):

P`(x) = 2 - (A/x^2)

Then set the equation to 0 find the extrema and to solve for the variable x.

0 = 2 - (A/x^2)

2 = A/(x^2)

x^2 = A/2

x = (A/2)^1/2

And then substituted x into the area equation to find y:

y((A/2)^1/2) = A

y = (A/(A/2)^1/2))

According to the answer key this is wrong and we've been unable to find our error. It seems like something in the setup is wrong but we've reached a block. Does anyone have any insight into this?

Thank you in advance!
 

Answers and Replies

  • #2
360
0
Welcome to PF!

Hm, everything looks okay as far as I can tell. Did you try simplifying your answer? I wonder if the book isn't just listing this simplified form since
[tex]
\frac{A}{\sqrt{\frac{A}{2}}}
[/tex]
can be simplified.
 
  • #3
I should have given the answer the book presented. Here it is:

x = ½√(2A)
y = √(2A)

I don't see how the solutions we got could be simplified into the solutions the book gives, however playing algebraically with roots is not my strongest skill. If this is just an algebraic situation I apologize in advance, but if anyone has some insight into this it is appreciated; I just want to make sure that our approach is sound.
 
  • #4
360
0
Try using some of the properties listed at http://en.wikipedia.org/wiki/Square_root#Properties to help simplify your answer. That should help you get the answer listed in the book. And in general, can you simplify
[tex]
\frac{a}{\frac{b}{c}} \; ?
[/tex]
 

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